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NCERT Exemplar · Class 9 Mathematics Quadrilaterals

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EXERCISE 8.1 1–10 (part 1 of 7)

  1. Write the correct answer in each of the following:

    Exercise 1

    Three angles of a quadrilateral are 75∘,90∘\displaystyle 75^{\circ}, 90^{\circ} and 75∘\displaystyle 75^{\circ}. The fourth angle is (A) 90\displaystyle 90° (B) 95\displaystyle 95° (C) 105\displaystyle 105° (D) 120\displaystyle 120°

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    NCERT’s answer
    (D)
    (D) \(\displaystyle 120^{\circ}\)\[\angle A+\angle B+\angle C+\angle D=360^{\circ} \quad \text{(angle sum of a quadrilateral)} \] \[75^{\circ}+90^{\circ}+75^{\circ}+\angle D=360^{\circ} \] \[\angle D=120^{\circ} \]

  2. Exercise 2

    A diagonal of a rectangle is inclined to one side of the rectangle at 25\displaystyle 25°. The acute angle between the diagonals is (A) 55\displaystyle 55° (B) 50\displaystyle 50° (C) 40\displaystyle 40° (D) 25\displaystyle 25°

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    NCERT’s answer
    (B)
    (B) \(\displaystyle 50^{\circ}\)NCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q2\[OA=OB \quad \text{(diagonals of a rectangle are equal and bisect each other)} \] \[\angle OAB=\angle OBA=25^{\circ} \quad \text{(isosceles }\triangle OAB\text{)} \] \[\angle AOB=180^{\circ}-25^{\circ}-25^{\circ}=130^{\circ} \] \[\angle AOD=180^{\circ}-130^{\circ}=50^{\circ} \]

  3. Exercise 3

    ABCD is a rhombus such that ∠ACB=40∘\displaystyle \angle \mathrm{ACB}=40^{\circ}. Then ∠ADB\displaystyle \angle \mathrm{ADB} is (A) 40\displaystyle 40° (B) 45\displaystyle 45° (C) 50\displaystyle 50° (D) 60\displaystyle 60°

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    NCERT’s answer
    (C)
    (C) \(\displaystyle 50^{\circ}\)NCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q3\[\angle ACB=\angle ACD=40^{\circ} \quad \text{(}AC\text{ bisects }\angle C\text{)} \] \[\angle C=80^{\circ},\quad \angle D=180^{\circ}-80^{\circ}=100^{\circ} \quad \text{(adjacent angles of a rhombus)} \] \[\angle ADB=\tfrac{1}{2}\angle D=50^{\circ} \quad \text{(}BD\text{ bisects }\angle D\text{)} \]

  4. Exercise 4

    The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if (A) PQRS is a rectangle (B) PQRS is a parallelogram (C) diagonals of PQRS are perpendicular (D) diagonals of PQRS are equal.

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    NCERT’s answer
    (C)
    (C) diagonals of PQRS are perpendicularNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q4\[WX\parallel PR,\quad XY\parallel QS \quad \text{(midpoint theorem)} \] WX and XY meet at a right angle only when PR is perpendicular to QS, making WXYZ a rectangle.

  5. Exercise 5

    The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rhombus, if (A) PQRS is a rhombus (B) PQRS is a parallelogram (C) diagonals of PQRS are perpendicular (D) diagonals of PQRS are equal.

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    NCERT’s answer
    (D)
    (D) diagonals of PQRS are equalNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q5\[WX=\tfrac12 PR,\quad XY=\tfrac12 QS \quad \text{(midpoint theorem)} \] WX and XY are equal only when the diagonals PR and QS are equal, making WXYZ a rhombus.

  6. Exercise 6

    If angles A,B,C\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{C} and D of the quadrilateral ABCD , taken in order, are in the ratio 3\displaystyle 3:7\displaystyle 7:6\displaystyle 6:4\displaystyle 4, then ABCD is a (A) rhombus (B) parallelogram (C) trapezium (D) kite

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    NCERT’s answer
    (C)
    (C) trapeziumNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q6\[3x+7x+6x+4x=360^{\circ} \quad \text{(angle sum of a quadrilateral)} \] \[20x=360^{\circ}\ \Rightarrow\ x=18^{\circ} \] \[\angle A=54^{\circ},\ \angle B=126^{\circ},\ \angle C=108^{\circ},\ \angle D=72^{\circ} \] \[\angle A+\angle B=180^{\circ} \quad \text{(co-interior angles)}\ \Rightarrow\ AD\parallel BC \]

  7. Exercise 7

    If bisectors of ∠A\displaystyle \angle \mathrm{A} and ∠B\displaystyle \angle \mathrm{B} of a quadrilateral ABCD intersect each other at P , of ∠B\displaystyle \angle \mathrm{B} and ∠C\displaystyle \angle \mathrm{C} at Q , of ∠C\displaystyle \angle \mathrm{C} and ∠D\displaystyle \angle \mathrm{D} at R and of ∠D\displaystyle \angle \mathrm{D} and ∠A\displaystyle \angle \mathrm{A} at S , then PQRS is a (A) rectangle (B) rhombus (C) parallelogram (D) quadrilateral whose opposite angles are supplementary

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    NCERT’s answer
    (D)
    (D) quadrilateral whose opposite angles are supplementaryNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q7\[\angle P=180^{\circ}-\tfrac12(\angle A+\angle B),\quad \angle R=180^{\circ}-\tfrac12(\angle C+\angle D) \quad \text{(angle sum of a triangle at }P,R\text{)} \] \[\angle P+\angle R=360^{\circ}-\tfrac12(\angle A+\angle B+\angle C+\angle D)=360^{\circ}-180^{\circ}=180^{\circ} \]

  8. Exercise 8

    If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form (A) a square (B) a rhombus (C) a rectangle (D) any other parallelogram

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    NCERT’s answer
    (C)
    (C) a rectangleNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q8\[\angle APQ+\angle BPQ=180^{\circ} \quad \text{(linear pair)} \] \[\angle CQP+\angle PQD=180^{\circ} \quad \text{(linear pair)} \] Halved, the two linear pairs give perpendicular bisectors at P and at Q; AB∥CD makes the opposite bisectors parallel, so PHQK is a rectangle, only a square in the special case PH=HQ.

  9. Exercise 9

    The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is (A) a rhombus (B) a rectangle (C) a square (D) any parallelogram

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    NCERT’s answer
    (B)
    (B) a rectangleNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q9\[AB=BC=CD=DA \quad \text{(sides of a rhombus)} \] \[WX\parallel AC,\quad XY\parallel BD \quad \text{(midpoint theorem)} \] \[AC\perp BD \quad \text{(diagonals of a rhombus)}\ \Rightarrow\ WX\perp XY \] WXYZ is a rectangle, and a square only when the rhombus is one too (its diagonals equal).

  10. Exercise 10

    D and E are the mid-points of the sides AB and AC of ΔABC\displaystyle \Delta \mathrm{ABC} and O is any point on side BC . O is joined to A . If P and Q are the mid-points of OB and OC respectively, then DEQP is (A) a square (B) a rectangle (C) a rhombus (D) a parallelogram

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    NCERT’s answer
    (D)
    (D) a parallelogramNCERT_Solution_Class9_Maths_Exemplar_Ch8_Ex8-1_Q10\[DE=\tfrac12 BC,\quad DE\parallel BC \quad \text{(}D,E\text{ midpoints of }AB,AC\text{)} \] \[PQ=\tfrac12 BC,\quad PQ\parallel BC \quad \text{(}P,Q\text{ midpoints of }OB,OC\text{)} \] \[DE=PQ,\quad DE\parallel PQ\ \Rightarrow\ DEQP\text{ is a parallelogram} \]